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b^d$ +- *Iterative Deepening Search* + - Frontier: Stack, DLS with max depth from $0..N$ + - *Time*: $O(b^(d))$ + - *Space*: $O(b d)$ + - *Completeness*: Yes + - *Optimality*: Yes if step cost is same. + - $N_(i d s) = (d+1)b^0 + (d)b^1 + ... + (1)b^(d)$ +- *Bidirectional Search* + - Search from initial state and goal state at same time since $b^(d/2) + b^(d/2) < b^d$. + +```python +frontier = []; initial_state = x +visited = set() # if graph search +frontier.add(initial_state) +while len(frontier) != 0: + state = frontier.pop() + if state in visited: continue # if graph search + visited.add(next_state) # if graph search + for action in actions(state): + next_state = transition(state, action) + if next_state == goal: return solution + frontier.add(next_state) +return failure +``` + +== Informed Search Algorithms +- *Greedy Best Fit Search* + - Frontier: Priority Queue(`f(n)`) + - *`f(n)`*: `h(n)`: Heuristic: estimated cost from `n` to goal. + - *Time*: $O(b^m)$, Good heuristic improves + - *Space*: $O(b^m)$ Keeps all nodes in memory + - *Complete*: No + - *Optimal*: No (Doesn't consider cost so far) +- *A\** + - Frontier: Priority Queue(`f(n)`) + - *`f(n)`*: `g(n) + h(n)`, `g` : total cost, `h` : estimated cost + - *Time*: $O(b^m)$, Good heuristic improves + - *Space*: $O(b^m)$ Keeps all nodes in memory + - *Complete*: Yes + - *Optimal*: Yes (Doesn't consider cost so far) +== Heuristics +=== Admissible +Admissible if for every node $n, h(n) lt.eq h\*(n)$, where $h\*(n)$ is the true cost to reach goal state from n. An admissible heuristic never over-estimates the cost to reach the goal, i.e. its a conservative estimate + +If $h(n)$ is admissible, A\* using *tree search* is optimal + +=== Consistent +If it obeys the triangle inequality, $forall n, h(n) lt.eq c(n, a, n') + h(n')$, or estimated cost of reaching goal node through node $n$ is $lt.eq$ to the estimated cost of reaching goal node through $n'$ + cost of going to node $n'$ from $n$. + +If our heuristic is consistent, then the first time we visit a node, the estimated cost to the goal $h(n)$ is guaranteed to be smallest. Therefore, we do not need to visit the node again as any other path we might visit it with has a larger overall cost, and graph search (i.e. tree search with memoisation) is optimal. +However, if our heuristic is not consistent, then consider a path that we might visit later where $h(n) gt.eq c(n, a, n') + h(n')$. Then, if we have visited $h(n)$ already, we still need to revisit it as we have found another shorter path later on in our traversal, hence making graph search sub-optimal. +=== Dominance +If $forall n, h_2(n) gt.eq h_1(n)$, then $h_2$ dominates $h_1$ and $h_2$ would be better for search. + +=== Creating Admissible Heuristics +Cost of an optimal solution to a relaxed problem (problem with fewer restrictions) is an admissible heuristic for the original problem + +== Local Search +When path to solution not important, state is the solution (chess, sudoku, bin packing) + +Keep the current state, and iteratively try improving using heuristic we define. Local search can get stuck in local minima/maxima, so random restarts can help get better results +=== Trivial Algorithms +- *Random Sampling* Random sample a state until solution is found +- *Random walk* Go to random neighbours until solution found + +=== Non-trivial Algorihthms +- *Hill Climbing* Pick the best among neighbours, repeat +- *Simulated Annealing* Hill climbing but allows bad moves +- *Beam Search* k-kill climbing in parallel +- *Genetic* Marry the best, mutate, repeat + +== Adversarial Search +When trying to find moves against rational agent. + +=== Minimax +heuristic $h(n)$ that defines "goodness" of current state. We want to _maximise_ $h(n)$, whereas opponent wants to _minimise_ $h(n)$. + +We can optimise minimax algo by introducing $alpha\/beta$ pruning, where $alpha$ is best value for max player, and $beta$ is best value for min player. If at any node, $alpha$ and $beta$ does not overlap($beta lt.eq alpha$), we can prune that node. + +- *Time*: $O(b^m)$ +- *Space*: $O(b m)$ +- *Complete* Yes, if tree is finite +- *Optimal* Yes, against optimal opponent + +#[ +#set text(size: 8pt) + +```python +def alpha_beta_search(state): + _ = max_value(state, -INF, INF) + return action in succesors(state) with value v + +def max_value(state, alpha, beta): + if is_terminal(state): return utility(state) + v = -INF + for action, next_state in successors(state): + min_v = min_value(next_state, alpha, beta) + v = max(v, min_v) + if v >= beta: return v + alpha = max(alpha, v) + return v + +def min_value(state, alpha, beta): + if is_terminal(state): return utility(state) + v = INF + for action, next_state in successors(state): + max_v = max_value(next_state, alpha, beta) + if v <= alpha: return v + beta = min(beta, v) + return v +``` +] + += Machine Learning and Decision Trees +== Supervised Learning +Learns from being given the right answers +- *Regression*: Predict Continuous outputs +- *Classification*: Predict discrete outputs + +Assumption: $y$ is generated by true mapping function $f: x arrow.r y$. We want to find a hypothesis $h:x arrow.r hat(y)$ + +== Performance Measure +We can measure error for regression: +- Absolute Error: $|hat(y) - y|$ +- Squared Error: $(hat(y) - y)^2$ + +For a set of $N$ examples, we can compute average error for regression: +- Mean Squared Error: $1/N sum^N_(i=1)(hat(y)_i - y_i)^2$ +- Mean Absolute Error: $1/N sum^N_(i=1)|hat(y)_i - y_i|$ + +Average correctness for classification: +$"Accuracy" = 1/N sum^N_(i=1)1_(hat(y)_i = y_i)$ + +Confusion Matrix: + +$"Accuracy" = ("TP" + "TN")/("TP" + "FN" + "FP" + "TN")$ + +$"Precision P" = ("TP")/("TP" + "FP")$ (Maximise if FP is very costly, e.g. email spam) + +$"Recall R" = ("TP")/("TP" + "FN")$ (Maximise if FN is bad, e.g. Cancer) + +$"F1" = 2/((1/P) + (1/R))$ (Maximise if FN is bad, e.g. Cancer) + +== Decision Trees +Choosing an attribute to split a decision tree: Ideally, select attribute that splits all examples into 2 distinct groups + +The amount of information at a given node is the entropy, $I(P(v_1), ..., P(v_n)) = -sum^n_(i=1)P(v_I)log_2P(v_i)$, where $v_i$ are the different classifications of the dataset. +For a binary classification, (Positive, Negative), $I(p/(p+n), n/(p+n)) = -p/(p+n)log_2 n/(p+n) - n/(p+n)log_2 n/(p+n)$ + +When an attribute is divided into subsets, we can calculate the _Information Gain_ (reduction in entropy) by + +$"IG"("Attrib") = I(P(p/(p+n)), P(n/(p+n))) - sum^v_(i=1)(p_i+n_i)/(p+n)I(p_i/(p_i+n_i), n_i/(p_i+n_i))$ + += Misc +== Minimax Intuition + +- In Maximiser: + - When $v >= alpha$, update #math.alpha. + - If $v >= beta$, prune the rest +- In Minimiser: + - When $v <= beta$, update #math.beta. + - If $v <= alpha$, prune the rest + +== Proof of Admissibility +- If a relaxed version of the problem's optimal solution is $H_a$, then $H_a$ is an admissible heuristic for the current problem +- Check if $h_a ("Goal") > 0$. If it is, then its inadmissible +- Compare cost of moves vs maximum rate of loss of heuristic. This is the intuition when checking consistency because if one move improves heuristic “too much”, it’s likely to be inconsistent. For example, in the snake question, if the snake can eat the apple and improve heuristic by 5 when the cost of the move to eat the apple is only 1, we can immediately say it’s inconsistent. +- Proof that it fails by counterexample / (contradiction / contrapositive) +- Proof that it passes by contradiction / contrapositive +- Proof that it passes by abuse of inequalities +- Proof that it passes by proving the correctness for extreme cases, and proving that everything in between is therefore correct (dodgy proof technique) +- Lemma that guarantees admissibility as long as we have consistency and h(Goal)=0 +- Contrapositive of this Lemma: inconsistent as long as inadmissible and h(Goal)=0 + + +== Proof of InAdmissibility +- Show that $exists n, H_a (n) > H\*(n)$ + +== Proof of Inconsistency +- First show that $H_a ("Goal") = 0$ +- Try show that $H_a (n) > c(n, a, n') + H_a(n')$, assuming $n'$ is goal, and $c$ is the true cost function. 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